Integrand size = 22, antiderivative size = 252 \[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx=\frac {\left (5+a+(-1+x)^2\right ) (-1+x)}{8 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^2}+\frac {\left ((6+a) (25+7 a)+6 (7+2 a) (-1+x)^2\right ) (-1+x)}{32 (3+a)^2 (4+a)^2 \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}-\frac {3 \left (80+7 a^2+14 \sqrt {4+a}+a \left (47+4 \sqrt {4+a}\right )\right ) \arctan \left (\frac {-1+x}{\sqrt {1-\sqrt {4+a}}}\right )}{64 (3+a)^2 (4+a)^{5/2} \sqrt {1-\sqrt {4+a}}}-\frac {3 \left (14+4 a-\frac {80+47 a+7 a^2}{\sqrt {4+a}}\right ) \arctan \left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )}{64 (3+a)^2 (4+a)^2 \sqrt {1+\sqrt {4+a}}} \]
1/8*(5+a+(-1+x)^2)*(-1+x)/(a^2+7*a+12)/(3+a-2*(-1+x)^2-(-1+x)^4)^2+1/32*(( 6+a)*(25+7*a)+6*(7+2*a)*(-1+x)^2)*(-1+x)/(a^2+7*a+12)^2/(3+a-2*(-1+x)^2-(- 1+x)^4)-3/64*arctan((-1+x)/(1-(4+a)^(1/2))^(1/2))*(80+7*a^2+14*(4+a)^(1/2) +a*(47+4*(4+a)^(1/2)))/(3+a)^2/(4+a)^(5/2)/(1-(4+a)^(1/2))^(1/2)-3/64*arct an((-1+x)/(1+(4+a)^(1/2))^(1/2))*(14+4*a+(-7*a^2-47*a-80)/(4+a)^(1/2))/(3+ a)^2/(4+a)^2/(1+(4+a)^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.09 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx=\frac {1}{128} \left (\frac {16 (-1+x) \left (6+a-2 x+x^2\right )}{(3+a) (4+a) \left (a-x \left (-8+8 x-4 x^2+x^3\right )\right )^2}+\frac {4 (-1+x) \left (7 a^2+6 \left (32-14 x+7 x^2\right )+a \left (79-24 x+12 x^2\right )\right )}{(3+a)^2 (4+a)^2 \left (a-x \left (-8+8 x-4 x^2+x^3\right )\right )}-\frac {3 \text {RootSum}\left [a+8 \text {$\#$1}-8 \text {$\#$1}^2+4 \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {108 \log (x-\text {$\#$1})+55 a \log (x-\text {$\#$1})+7 a^2 \log (x-\text {$\#$1})-28 \log (x-\text {$\#$1}) \text {$\#$1}-8 a \log (x-\text {$\#$1}) \text {$\#$1}+14 \log (x-\text {$\#$1}) \text {$\#$1}^2+4 a \log (x-\text {$\#$1}) \text {$\#$1}^2}{-2+4 \text {$\#$1}-3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ]}{\left (12+7 a+a^2\right )^2}\right ) \]
((16*(-1 + x)*(6 + a - 2*x + x^2))/((3 + a)*(4 + a)*(a - x*(-8 + 8*x - 4*x ^2 + x^3))^2) + (4*(-1 + x)*(7*a^2 + 6*(32 - 14*x + 7*x^2) + a*(79 - 24*x + 12*x^2)))/((3 + a)^2*(4 + a)^2*(a - x*(-8 + 8*x - 4*x^2 + x^3))) - (3*Ro otSum[a + 8*#1 - 8*#1^2 + 4*#1^3 - #1^4 & , (108*Log[x - #1] + 55*a*Log[x - #1] + 7*a^2*Log[x - #1] - 28*Log[x - #1]*#1 - 8*a*Log[x - #1]*#1 + 14*Lo g[x - #1]*#1^2 + 4*a*Log[x - #1]*#1^2)/(-2 + 4*#1 - 3*#1^2 + #1^3) & ])/(1 2 + 7*a + a^2)^2)/128
Time = 0.44 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2458, 1405, 27, 1492, 27, 1480, 217}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\left (a-x^4+4 x^3-8 x^2+8 x\right )^3} \, dx\) |
| \(\Big \downarrow \) 2458 |
| \(\displaystyle \int \frac {1}{\left (a-(x-1)^4-2 (x-1)^2+3\right )^3}d(x-1)\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}-\frac {\int -\frac {2 \left (5 (x-1)^2+7 a+27\right )}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)}{16 \left (a^2+7 a+12\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 (x-1)^2+7 a+27}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)}{8 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {\frac {(x-1) \left (6 (2 a+7) (x-1)^2+(a+6) (7 a+25)\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\int -\frac {6 \left (7 a^2+51 a+2 (2 a+7) (x-1)^2+94\right )}{-(x-1)^4-2 (x-1)^2+a+3}d(x-1)}{8 \left (a^2+7 a+12\right )}}{8 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {7 a^2+51 a+2 (2 a+7) (x-1)^2+94}{-(x-1)^4-2 (x-1)^2+a+3}d(x-1)}{4 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (6 (2 a+7) (x-1)^2+(a+6) (7 a+25)\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}}{8 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {3 \left (\frac {1}{2} \left (-\frac {7 a^2+47 a+80}{\sqrt {a+4}}+4 a+14\right ) \int \frac {1}{-(x-1)^2-\sqrt {a+4}-1}d(x-1)+\frac {1}{2} \left (\frac {7 a^2+47 a+80}{\sqrt {a+4}}+4 a+14\right ) \int \frac {1}{-(x-1)^2+\sqrt {a+4}-1}d(x-1)\right )}{4 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (6 (2 a+7) (x-1)^2+(a+6) (7 a+25)\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}}{8 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {\left (\frac {7 a^2+47 a+80}{\sqrt {a+4}}+4 a+14\right ) \arctan \left (\frac {x-1}{\sqrt {1-\sqrt {a+4}}}\right )}{2 \sqrt {1-\sqrt {a+4}}}-\frac {\left (-\frac {7 a^2+47 a+80}{\sqrt {a+4}}+4 a+14\right ) \arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )}{2 \sqrt {\sqrt {a+4}+1}}\right )}{4 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (6 (2 a+7) (x-1)^2+(a+6) (7 a+25)\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}}{8 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}\) |
((5 + a + (-1 + x)^2)*(-1 + x))/(8*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^2) + ((((6 + a)*(25 + 7*a) + 6*(7 + 2*a)*(-1 + x)^2)*(-1 + x ))/(4*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (3*(-1/2*((1 4 + 4*a + (80 + 47*a + 7*a^2)/Sqrt[4 + a])*ArcTan[(-1 + x)/Sqrt[1 - Sqrt[4 + a]]])/Sqrt[1 - Sqrt[4 + a]] - ((14 + 4*a - (80 + 47*a + 7*a^2)/Sqrt[4 + a])*ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]])/(2*Sqrt[1 + Sqrt[4 + a]])))/( 4*(12 + 7*a + a^2)))/(8*(12 + 7*a + a^2))
3.2.22.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) ), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.59
| method | result | size |
| default | \(-\frac {\frac {3 \left (7+2 a \right ) x^{7}}{16 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}-\frac {21 \left (7+2 a \right ) x^{6}}{16 \left (a^{2}+8 a +16\right ) \left (a^{2}+6 a +9\right )}+\frac {\left (7 a^{2}+343 a +1116\right ) x^{5}}{32 a^{4}+448 a^{3}+2336 a^{2}+5376 a +4608}-\frac {5 \left (7 a^{2}+175 a +528\right ) x^{4}}{32 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}+\frac {\left (34 a^{2}+679 a +1968\right ) x^{3}}{16 a^{4}+224 a^{3}+1168 a^{2}+2688 a +2304}-\frac {\left (32 a^{2}+623 a +1800\right ) x^{2}}{16 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}-\frac {\left (11 a^{3}+107 a^{2}-84 a -1152\right ) x}{32 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}+\frac {11 a^{3}+131 a^{2}+408 a +288}{32 \left (3+a \right ) \left (a^{3}+11 a^{2}+40 a +48\right )}}{\left (-x^{4}+4 x^{3}-8 x^{2}+a +8 x \right )^{2}}-\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )}{\sum }\frac {\left (-108+2 \left (-2 a -7\right ) \textit {\_R}^{2}+4 \left (7+2 a \right ) \textit {\_R} -7 a^{2}-55 a \right ) \ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{3}+3 \textit {\_R}^{2}-4 \textit {\_R} +2}\right )}{128 \left (a^{3}+10 a^{2}+33 a +36\right ) \left (4+a \right )}\) | \(400\) |
| risch | \(\frac {-\frac {3 \left (7+2 a \right ) x^{7}}{16 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}+\frac {21 \left (7+2 a \right ) x^{6}}{16 \left (a^{2}+8 a +16\right ) \left (a^{2}+6 a +9\right )}-\frac {\left (7 a^{2}+343 a +1116\right ) x^{5}}{32 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}+\frac {5 \left (7 a^{2}+175 a +528\right ) x^{4}}{32 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}-\frac {\left (34 a^{2}+679 a +1968\right ) x^{3}}{16 \left (a^{4}+14 a^{3}+73 a^{2}+168 a +144\right )}+\frac {\left (32 a^{2}+623 a +1800\right ) x^{2}}{16 a^{4}+224 a^{3}+1168 a^{2}+2688 a +2304}+\frac {\left (11 a^{3}+107 a^{2}-84 a -1152\right ) x}{32 a^{4}+448 a^{3}+2336 a^{2}+5376 a +4608}-\frac {11 a^{3}+131 a^{2}+408 a +288}{32 \left (3+a \right ) \left (a^{3}+11 a^{2}+40 a +48\right )}}{\left (-x^{4}+4 x^{3}-8 x^{2}+a +8 x \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )}{\sum }\frac {\left (\frac {2 \left (7+2 a \right ) \textit {\_R}^{2}}{a^{4}+14 a^{3}+73 a^{2}+168 a +144}-\frac {4 \left (7+2 a \right ) \textit {\_R}}{a^{4}+14 a^{3}+73 a^{2}+168 a +144}+\frac {7 a +27}{a^{3}+10 a^{2}+33 a +36}\right ) \ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{3}+3 \textit {\_R}^{2}-4 \textit {\_R} +2}\right )}{128}\) | \(431\) |
-(3/16*(7+2*a)/(a^4+14*a^3+73*a^2+168*a+144)*x^7-21/16*(7+2*a)/(a^2+8*a+16 )/(a^2+6*a+9)*x^6+1/32*(7*a^2+343*a+1116)/(a^4+14*a^3+73*a^2+168*a+144)*x^ 5-5/32*(7*a^2+175*a+528)/(a^4+14*a^3+73*a^2+168*a+144)*x^4+1/16*(34*a^2+67 9*a+1968)/(a^4+14*a^3+73*a^2+168*a+144)*x^3-1/16*(32*a^2+623*a+1800)/(a^4+ 14*a^3+73*a^2+168*a+144)*x^2-1/32*(11*a^3+107*a^2-84*a-1152)/(a^4+14*a^3+7 3*a^2+168*a+144)*x+1/32*(11*a^3+131*a^2+408*a+288)/(3+a)/(a^3+11*a^2+40*a+ 48))/(-x^4+4*x^3-8*x^2+a+8*x)^2-3/128/(a^3+10*a^2+33*a+36)/(4+a)*sum((-108 +2*(-2*a-7)*_R^2+4*(7+2*a)*_R-7*a^2-55*a)/(-_R^3+3*_R^2-4*_R+2)*ln(x-_R),_ R=RootOf(_Z^4-4*_Z^3+8*_Z^2-8*_Z-a))
Leaf count of result is larger than twice the leaf count of optimal. 3971 vs. \(2 (220) = 440\).
Time = 0.31 (sec) , antiderivative size = 3971, normalized size of antiderivative = 15.76 \[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx=\text {Too large to display} \]
-1/128*(24*(2*a + 7)*x^7 - 168*(2*a + 7)*x^6 + 4*(7*a^2 + 343*a + 1116)*x^ 5 - 20*(7*a^2 + 175*a + 528)*x^4 + 8*(34*a^2 + 679*a + 1968)*x^3 + 44*a^3 - 8*(32*a^2 + 623*a + 1800)*x^2 - 3*((a^4 + 14*a^3 + 73*a^2 + 168*a + 144) *x^8 - 8*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^7 + 32*(a^4 + 14*a^3 + 73 *a^2 + 168*a + 144)*x^6 + a^6 - 80*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x ^5 + 14*a^5 - 2*(a^5 - 50*a^4 - 823*a^3 - 4504*a^2 - 10608*a - 9216)*x^4 + 73*a^4 + 8*(a^5 - 2*a^4 - 151*a^3 - 1000*a^2 - 2544*a - 2304)*x^3 + 168*a ^3 - 16*(a^5 + 10*a^4 + 17*a^3 - 124*a^2 - 528*a - 576)*x^2 + 144*a^2 + 16 *(a^5 + 14*a^4 + 73*a^3 + 168*a^2 + 144*a)*x)*sqrt((105*a^4 + 1470*a^3 + 7 749*a^2 + (a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a^5 + 3 73020*a^4 + 735840*a^3 + 950400*a^2 + 725760*a + 248832)*sqrt((2401*a^4 + 33124*a^3 + 171966*a^2 + 398164*a + 346921)/(a^15 + 50*a^14 + 1165*a^13 + 16780*a^12 + 167090*a^11 + 1218460*a^10 + 6722130*a^9 + 28570320*a^8 + 943 20045*a^7 + 241870050*a^6 + 477857313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 277136640*a + 60466176)) + 18228*a + 16144)/(a^10 + 35*a ^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a^5 + 373020*a^4 + 735840*a^3 + 950400*a^2 + 725760*a + 248832))*log(-64827*a^4 - 907578*a^3 - 4780647* a^2 + 27*(2401*a^4 + 33614*a^3 + 177061*a^2 + 415884*a + 367536)*x + 27*(3 43*a^7 + 8981*a^6 + 100811*a^5 + 628887*a^4 + 2354874*a^3 + 5293208*a^2 - (11*a^12 + 462*a^11 + 8881*a^10 + 103320*a^9 + 810205*a^8 + 4511542*a^7...
Leaf count of result is larger than twice the leaf count of optimal. 697 vs. \(2 (230) = 460\).
Time = 8.02 (sec) , antiderivative size = 697, normalized size of antiderivative = 2.77 \[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx=- \frac {11 a^{3} + 131 a^{2} + 408 a + x^{7} \cdot \left (12 a + 42\right ) + x^{6} \left (- 84 a - 294\right ) + x^{5} \cdot \left (7 a^{2} + 343 a + 1116\right ) + x^{4} \left (- 35 a^{2} - 875 a - 2640\right ) + x^{3} \cdot \left (68 a^{2} + 1358 a + 3936\right ) + x^{2} \left (- 64 a^{2} - 1246 a - 3600\right ) + x \left (- 11 a^{3} - 107 a^{2} + 84 a + 1152\right ) + 288}{32 a^{6} + 448 a^{5} + 2336 a^{4} + 5376 a^{3} + 4608 a^{2} + x^{8} \cdot \left (32 a^{4} + 448 a^{3} + 2336 a^{2} + 5376 a + 4608\right ) + x^{7} \left (- 256 a^{4} - 3584 a^{3} - 18688 a^{2} - 43008 a - 36864\right ) + x^{6} \cdot \left (1024 a^{4} + 14336 a^{3} + 74752 a^{2} + 172032 a + 147456\right ) + x^{5} \left (- 2560 a^{4} - 35840 a^{3} - 186880 a^{2} - 430080 a - 368640\right ) + x^{4} \left (- 64 a^{5} + 3200 a^{4} + 52672 a^{3} + 288256 a^{2} + 678912 a + 589824\right ) + x^{3} \cdot \left (256 a^{5} - 512 a^{4} - 38656 a^{3} - 256000 a^{2} - 651264 a - 589824\right ) + x^{2} \left (- 512 a^{5} - 5120 a^{4} - 8704 a^{3} + 63488 a^{2} + 270336 a + 294912\right ) + x \left (512 a^{5} + 7168 a^{4} + 37376 a^{3} + 86016 a^{2} + 73728 a\right )} - \operatorname {RootSum} {\left (t^{4} \cdot \left (268435456 a^{15} + 14763950080 a^{14} + 378493992960 a^{13} + 5999532441600 a^{12} + 65757291479040 a^{11} + 527875908304896 a^{10} + 3206246773555200 a^{9} + 15003759578972160 a^{8} + 54537151127224320 a^{7} + 153980418717122560 a^{6} + 334927734494986240 a^{5} + 551152193655275520 a^{4} + 664192984106926080 a^{3} + 553362212027105280 a^{2} + 284993413919539200 a + 68398419340689408\right ) + t^{2} \left (- 30965760 a^{9} - 1052835840 a^{8} - 15910207488 a^{7} - 140262506496 a^{6} - 795007254528 a^{5} - 3004516270080 a^{4} - 7571263979520 a^{3} - 12268037210112 a^{2} - 11598827618304 a - 4875324751872\right ) - 194481 a^{4} - 2762424 a^{3} - 14762736 a^{2} - 35178624 a - 31539456, \left ( t \mapsto t \log {\left (x + \frac {23068672 t^{3} a^{12} + 968884224 t^{3} a^{11} + 18624806912 t^{3} a^{10} + 216677744640 t^{3} a^{9} + 1699123036160 t^{3} a^{8} + 9461389328384 t^{3} a^{7} + 38361186172928 t^{3} a^{6} + 114107491549184 t^{3} a^{5} + 247138458009600 t^{3} a^{4} + 380084473036800 t^{3} a^{3} + 394002582994944 t^{3} a^{2} + 247177515368448 t^{3} a + 70970039599104 t^{3} - 395136 t a^{7} - 11676672 t a^{6} - 144076032 t a^{5} - 969518592 t a^{4} - 3861475200 t a^{3} - 9133300224 t a^{2} - 11906574336 t a - 6611337216 t - 64827 a^{4} - 907578 a^{3} - 4780647 a^{2} - 11228868 a - 9923472}{64827 a^{4} + 907578 a^{3} + 4780647 a^{2} + 11228868 a + 9923472} \right )} \right )\right )} \]
-(11*a**3 + 131*a**2 + 408*a + x**7*(12*a + 42) + x**6*(-84*a - 294) + x** 5*(7*a**2 + 343*a + 1116) + x**4*(-35*a**2 - 875*a - 2640) + x**3*(68*a**2 + 1358*a + 3936) + x**2*(-64*a**2 - 1246*a - 3600) + x*(-11*a**3 - 107*a* *2 + 84*a + 1152) + 288)/(32*a**6 + 448*a**5 + 2336*a**4 + 5376*a**3 + 460 8*a**2 + x**8*(32*a**4 + 448*a**3 + 2336*a**2 + 5376*a + 4608) + x**7*(-25 6*a**4 - 3584*a**3 - 18688*a**2 - 43008*a - 36864) + x**6*(1024*a**4 + 143 36*a**3 + 74752*a**2 + 172032*a + 147456) + x**5*(-2560*a**4 - 35840*a**3 - 186880*a**2 - 430080*a - 368640) + x**4*(-64*a**5 + 3200*a**4 + 52672*a* *3 + 288256*a**2 + 678912*a + 589824) + x**3*(256*a**5 - 512*a**4 - 38656* a**3 - 256000*a**2 - 651264*a - 589824) + x**2*(-512*a**5 - 5120*a**4 - 87 04*a**3 + 63488*a**2 + 270336*a + 294912) + x*(512*a**5 + 7168*a**4 + 3737 6*a**3 + 86016*a**2 + 73728*a)) - RootSum(_t**4*(268435456*a**15 + 1476395 0080*a**14 + 378493992960*a**13 + 5999532441600*a**12 + 65757291479040*a** 11 + 527875908304896*a**10 + 3206246773555200*a**9 + 15003759578972160*a** 8 + 54537151127224320*a**7 + 153980418717122560*a**6 + 334927734494986240* a**5 + 551152193655275520*a**4 + 664192984106926080*a**3 + 553362212027105 280*a**2 + 284993413919539200*a + 68398419340689408) + _t**2*(-30965760*a* *9 - 1052835840*a**8 - 15910207488*a**7 - 140262506496*a**6 - 795007254528 *a**5 - 3004516270080*a**4 - 7571263979520*a**3 - 12268037210112*a**2 - 11 598827618304*a - 4875324751872) - 194481*a**4 - 2762424*a**3 - 14762736...
\[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx=\int { -\frac {1}{{\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x\right )}^{3}} \,d x } \]
-1/32*(6*(2*a + 7)*x^7 - 42*(2*a + 7)*x^6 + (7*a^2 + 343*a + 1116)*x^5 - 5 *(7*a^2 + 175*a + 528)*x^4 + 2*(34*a^2 + 679*a + 1968)*x^3 + 11*a^3 - 2*(3 2*a^2 + 623*a + 1800)*x^2 + 131*a^2 - (11*a^3 + 107*a^2 - 84*a - 1152)*x + 408*a + 288)/((a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^8 - 8*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^7 + 32*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^ 6 + a^6 - 80*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^5 + 14*a^5 - 2*(a^5 - 50*a^4 - 823*a^3 - 4504*a^2 - 10608*a - 9216)*x^4 + 73*a^4 + 8*(a^5 - 2*a ^4 - 151*a^3 - 1000*a^2 - 2544*a - 2304)*x^3 + 168*a^3 - 16*(a^5 + 10*a^4 + 17*a^3 - 124*a^2 - 528*a - 576)*x^2 + 144*a^2 + 16*(a^5 + 14*a^4 + 73*a^ 3 + 168*a^2 + 144*a)*x) - 3/32*integrate((2*(2*a + 7)*x^2 + 7*a^2 - 4*(2*a + 7)*x + 55*a + 108)/(x^4 - 4*x^3 + 8*x^2 - a - 8*x), x)/(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)
Leaf count of result is larger than twice the leaf count of optimal. 16632 vs. \(2 (220) = 440\).
Time = 11.96 (sec) , antiderivative size = 16632, normalized size of antiderivative = 66.00 \[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx=\text {Too large to display} \]
-3/128*(sqrt((105*a^5 + 1890*a^4 + 13629*a^3 + 49224*a^2 + (49*a^5 + 926*a ^4 + 6997*a^3 + 26428*a^2 + 49904*a + 37696)*sqrt(a + 4) + 89056*a + 64576 )/(a^3 + 11*a^2 + 40*a + 48))*log(abs(16807*sqrt(a + 4)*a^15 + 26411*a^15 + 908950*sqrt(a + 4)*a^14 + 1420804*a^14 + 22929088*sqrt(a + 4)*a^13 + 240 1*sqrt(105*a^6 + 2205*a^5 + 19299*a^4 + 90111*a^3 + 236728*a^2 + (49*a^6 + 1073*a^5 + 9775*a^4 + 47419*a^3 + 129188*a^2 + 187408*a + 113088)*sqrt(a + 4) + 331744*a + 193728)*a^12*x + 35650176*a^13 + 2401*sqrt(105*a^6 + 220 5*a^5 + 19299*a^4 + 90111*a^3 + 236728*a^2 + (49*a^6 + 1073*a^5 + 9775*a^4 + 47419*a^3 + 129188*a^2 + 187408*a + 113088)*sqrt(a + 4) + 331744*a + 19 3728)*sqrt(a + 4)*a^11*x - 2401*sqrt(105*a^6 + 2205*a^5 + 19299*a^4 + 9011 1*a^3 + 236728*a^2 + (49*a^6 + 1073*a^5 + 9775*a^4 + 47419*a^3 + 129188*a^ 2 + 187408*a + 113088)*sqrt(a + 4) + 331744*a + 193728)*a^12 + 357887692*s qrt(a + 4)*a^12 + 105154*sqrt(105*a^6 + 2205*a^5 + 19299*a^4 + 90111*a^3 + 236728*a^2 + (49*a^6 + 1073*a^5 + 9775*a^4 + 47419*a^3 + 129188*a^2 + 187 408*a + 113088)*sqrt(a + 4) + 331744*a + 193728)*a^11*x - 2401*sqrt(105*a^ 6 + 2205*a^5 + 19299*a^4 + 90111*a^3 + 236728*a^2 + (49*a^6 + 1073*a^5 + 9 775*a^4 + 47419*a^3 + 129188*a^2 + 187408*a + 113088)*sqrt(a + 4) + 331744 *a + 193728)*sqrt(a + 4)*a^11 + 553458148*a^12 + 95550*sqrt(105*a^6 + 2205 *a^5 + 19299*a^4 + 90111*a^3 + 236728*a^2 + (49*a^6 + 1073*a^5 + 9775*a^4 + 47419*a^3 + 129188*a^2 + 187408*a + 113088)*sqrt(a + 4) + 331744*a + ...
Time = 12.12 (sec) , antiderivative size = 8242, normalized size of antiderivative = 32.71 \[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx=\text {Too large to display} \]
atan(((((52357496832*a + 57139003392*a^2 + 36322148352*a^3 + 14822473728*a ^4 + 4027170816*a^5 + 728506368*a^6 + 84615168*a^7 + 5726208*a^8 + 172032* a^9 + 21290287104)/(16384*(940032*a + 1195776*a^2 + 899328*a^3 + 442864*a^ 4 + 149208*a^5 + 34833*a^6 + 5564*a^7 + 582*a^8 + 36*a^9 + a^10 + 331776)) + ((4290672328704*a + 6001143054336*a^2 + 5025917042688*a^3 + 28005200035 84*a^4 + 1090200272896*a^5 + 302556119040*a^6 + 59862155264*a^7 + 82753617 92*a^8 + 761266176*a^9 + 41943040*a^10 + 1048576*a^11 + 1391569403904)/(16 384*(940032*a + 1195776*a^2 + 899328*a^3 + 442864*a^4 + 149208*a^5 + 34833 *a^6 + 5564*a^7 + 582*a^8 + 36*a^9 + a^10 + 331776)) - (x*(3510632448*a + 4020240384*a^2 + 2678587392*a^3 + 1144324096*a^4 + 325074944*a^5 + 6140723 2*a^6 + 7438336*a^7 + 524288*a^8 + 16384*a^9 + 1358954496))/(256*(48384*a + 49248*a^2 + 28560*a^3 + 10321*a^4 + 2380*a^5 + 342*a^6 + 28*a^7 + a^8 + 20736)))*((9*(39329792*a - 338*a*((a + 4)^15)^(1/2) - 589*((a + 4)^15)^(1/ 2) - 49*a^2*((a + 4)^15)^(1/2) + 41598976*a^2 + 25672960*a^3 + 10187840*a^ 4 + 2695744*a^5 + 475608*a^6 + 53949*a^7 + 3570*a^8 + 105*a^9 + 16531456)) /(16384*(1061683200*a + 2061434880*a^2 + 2474311680*a^3 + 2053201920*a^4 + 1247703040*a^5 + 573621760*a^6 + 203166720*a^7 + 55893360*a^8 + 11944200* a^9 + 1966491*a^10 + 244965*a^11 + 22350*a^12 + 1410*a^13 + 55*a^14 + a^15 + 254803968)))^(1/2))*((9*(39329792*a - 338*a*((a + 4)^15)^(1/2) - 589*(( a + 4)^15)^(1/2) - 49*a^2*((a + 4)^15)^(1/2) + 41598976*a^2 + 25672960*...